chemical reaction and radiation effects on unsteady mhd ... · pdf filemhd free convection...
TRANSCRIPT
Global Journal of Pure and Applied Mathematics.
ISSN 0973-1768 Volume 13, Number 7 (2017), pp. 3203-3236
Research India Publications
http://www.ripublication.com
Chemical Reaction and Radiation Effects on Unsteady
MHD Free Convection Flow Past an Exponentially
Accelerated Vertical Plate Through Porous Medium
With Heat Absorption in the Presence of Thermal
Diffusion
P. Gurivi Reddy1 T. Sudhakar Reddy2 S. V. K. Varma2
1Reader in mathematics, S. B. V. R. Degree college, Badvel, A.P., India. 2Department Mathematics, S. V. University, Tirupati, A. P. India.
Corresponding author
Abstract
The present study is aimed at analyzing the effects chemical reaction and
thermal radiation on an unsteady MHD free convection flow near a moving
vertical plate through a porous medium in presence of heat generation/
absorption. General exact solutions for the partial differential equations
governing the flow are obtained with the aid of the usual Laplace transform
technique. Also the applications of general solution for the important cases of
the flow are discussed. Velocity decreases with the increase in the magnetic
parameter or the radiation parameter , the coefficient of heat absorption , the
Schmidt number or Prandtl number. The skin friction coefficient increased
due to increase in the concentration buoyancy effects while it decreases due to
increase the magnetic parameter. The Nusselt number and Sherwood number
increases with increasing values of radiation parameter or heat absorption
parameter or chemical reaction parameter.
Key wards: MHD, Heat generation/ absorption, Radiation, Chemical reaction,
Porous medium, Laplace Transform method.
1 INTRODUCTION:
Gupta et al. [1] studied the viscous dissipation effects on free convection on flow past
a linearly accelerated vertical plate in the presence of viscous dissipative heat using
perturbation method. Kafousias and Raptis [2] extended this problem to include mass
transfer effects subjected to variable suction or injection. Free convection effects on
mailto:[email protected]
3204 P. Gurivi Reddy, T. Sudhakar Reddy and S.V.K. Varma
flow past an exponentially accelerated vertical plate was investigated by Singh and
Naveen kumar [3]. Soundalgekar et al. [4] considered Mass transfer effects on the
flow past an impulsively started infinite vertical plate with variable temperature
constant heat flux. The skin friction for accelerated vertical plate has been studied
analytically by Hossain and Shayo [5]. Basant kumar Jha et al. [6] analyzed mass
transfer effects on exponentially accelerated infinite vertical plate with constant heat
flux and uniform mass diffusion. Basant kumar Jha [7] studied MHD free convection
and mass transform flow through a porous medium. Muthucumaraswamy et al. [10]
studied mass transfer effects on exponentially accelerated isothermal vertical plate.
Radiation and mass transfer effects on MHD free convection flow past an
exponentially accelerated vertical plate with variable temperature was investigated by
Rajesh and Varma [11]. MHD effects on free convection and mass transform flow
through a porous medium with variable temperature was investigated by Rajesh and
Varma [12]. Muthucumaraswamy and Muralidharan [13] studied Thermal radiation
on linearly accelerated vertical plate with variable temperature and uniform mass flux.
Rajput U. S and Sahu P.S [14] investigated the effects of rotation and magnetic field
on the flow past an exponentially accelerated vertical plate with constant temperature.
Muthucumaraswamy and Visalakshi [15] studied radiative flow past an exponentially
accelerated vertical plate with variable temperature and mass diffusion. Recently
Rajput and Kumar [16] investigated radiation effects on MHD flow past an
impulsively started vertical plate with variable heat and mass transfer. Several
researchers reported on these issues [17-35]. The present study is aimed at analyzing
the effects chemical reaction and radiation on unsteady MHD free convection flow
near a moving vertical plate through a porous medium in presence of heat
generation/ absorption. A general exact solution for the partial differential equations
governing the flow is obtained with the aid of the usual Laplace transform technique.
Also the applications of general solution for the important cases of the flow are
discussed.
2 NOMENCLATURE
a* : Acceleration coefficient qr : Radiative heat flux in the y direction
A : Constant : Dimension less time
B0 : External magnetic field *t : Time *C : Specifies concentration in the Fluid t :Dimensionless time *
wC : Fluid concentration of the plate 0S : Soret number *C : Fluid concentration far away from the
plate
*y : Co-ordinate axis normal to the plate
C : Dimensionless concentration y : Dimensionless Co-ordinate axis normal to the plate
Chemical Reaction and Radiation Effects on Unsteady MHD Free Convection 3205
pC : Specifies heat at constant pressure GREEK SYMBOLS
H : Heat absorption Parameter : Thermal conductivity of the fluid D : Chemical molecular Diffusivity : Kinematic viscosity
Gm : mass Grashof number : Electrical conductivity
Gr : thermal Grashof number : Dynamic viscosity
g : gravitation due to acceleration T :coefficient of volume expansion
k : non dimensional permeability
coefficient of a porous medium
: thermal conductivity
CK : Non dimensional rate of chemical reaction
c : coefficient of volume expansion with
concentration
*
cK : Rate of chemical reaction : fluid density
k* : permeability of porous medium : non-dimensional skin friction
Nu : Nusselt number erf :Error function
Pr : Prandtl number erfc : Complementary error function
Sc : Schmidt number
SUBSCRIPTS AND SUPER SCRIPTS
*T : Temperature fluid near the plate W : plate *
wT : Temperature of the plate : for away from the plate
*T : Temperature of the fluid far away from the plate
Prime denotes differentiation with respect to y.
u*, v* : velocity components]
u : non dimensional velocity
F : radiation parameter
M : magnetic parameter
Sh : Non dimensional Sherwood number
5.3 FORMULATION OF THE PROBLEM:
We have considered the unsteady flow of an incompressible and electrically
conducting viscous fluid past an infinite vertical plate with variable temperature
embedded in porous medium in presence of chemical reaction and heat absorption. A
magnetic field of uniform strength 0B is applied transversely to the plate. The induced
magnetic field is neglected as the magnetic Reynolds number of the flow is taken to
be very small. The viscous dissipation is assumed to be negligible. The flow is
assumed to be in *x -direction which is taken along the vertical plate in the upward direction. The *y -axis is taken to be normal to the plate. Initially the plate and the
fluid are at the same temperature *T in the stationary condition with concentration
HCL 3Sticky NotePlease correct and update all correctionsin given word file from our side
3206 P. Gurivi Reddy, T. Sudhakar Reddy and S.V.K. Varma
level *C at all points. At time * 0t the plate is exponentially accelerated with a
velocity * *
0
a tu u e in its own plane and the plate temperature is raised linearly with time t and the level of concentration near the plate is raised to
*
wC . The fluid considered here is a gray, absorbing/emitting radiation but a non-scattering medium.
In this analysis we made the following assumptions:
1. It is assumed that there is no applied voltage which implies the absence of an electrical field.
2. The transversely applied magnetic field and magnetic Reynolds number are assumed to be very small so that the induced magnetic field and the Hall
Effect are negligible.
3. Viscous dissipation and Joule heating terms are neglected. 4. As the plate is infinite in extent, the physical variables are functions of y and
t only.
5. It is assumed that the effect of viscous dissipation is negligible in the energy equation
6. There is a first order chemical reaction between the diffusing species and the fluid.
7. The fluid considered here is a gray, absorbing emitting radiation but a non-scattering medium.
8. Thermo diffusion effect is considered.
Using the above assumptions and the usual Bossinesqs approximation, the unsteady
flow is governed by the following equations:
Momentum equation:
2
2* ** * * * *
2*0
* **( ) ( )T C
Bu uv g T T g C C u ut y k
(5.1)
Energy equation:
2* 2 *
* *
*
*
*1 r
P P P
qT Tt C C y
Q TCy
T
(5.2)
Concentration equation:
2 2
* 2 * 2 *
1
**
** *
*( )cC C TD Dt
Cy
Cy
K
(5.3)
Chemical Reaction and Radiation Effects on Unsteady MHD Free Convection 3207
The initial and boundary conditions are
* * * * * * *0, , , 0u